Detection and characterization of changes of the correlation structure in multivariate time series

We propose a method based on the equal-time correlation matrix as a sensitive detector for phase-shape correlations in multivariate data sets. The key point of the method is that changes of the degree of synchronization between time series provoke level repulsions between eigenstates at both edges of the spectrum of the correlation matrix. Consequently, detailed information about the correlation structure of the multivariate data set is imprinted into the dynamics of the eigenvalues and into the structure of the corresponding eigenvectors. The performance of the technique is demonstrated by application to $N_f$-tori, autoregressive models, and coupled chaotic systems. The high sensitivity, the comparatively small computational effort, and the excellent time resolution of the method recommend it for application to the analysis of complex, spatially extended, nonstationary systems.

Figure 1

Time evolution of (a) the largest eigenvalue ${\lambda }_{20}$, (b) the standard deviation of the phase distribution ${\sigma }_{\delta }$, (c) number of principal components $N_{20}^p$, and (d) the symmetry parameter $S_{20}$ of the largest eigenvector ${\vec{v}}_{20}$. The figure displays the results for systems $A$ (dotted line), $B$ (dashed line), and $C$ (solid line) of Table . The time scale is given in arbitrary units as explained in the text.

Figure 2

Results for system $D$ of Table . (a) Largest eigenvalue ${\lambda }_{20}$, (b) number of principal components $N_{20}^p$, and (c) the symmetry parameter $S_{20}$ of the largest eigenvector ${\vec{v}}_{20}$. (d), (e), and (f) show the corresponding quantities for the second smallest state and (g), (h), and (i) for the smallest state. The time scale is given in arbitrary units as explained in the text.

Figure 3

Distribution of the eigenvector components ${\mid a_{ij}\mid }^2$ for the smallest eigenvector (a) and (b) and for the largest eigenvector (c) and (d). The results in (a) and (c) were calculated for the first segment of system $D$ of Table , while (b) and (d) show the results of the last, correlated segment of system $D$. The solid line is obtained by averaging over 500 different realizations of system $D$. Dashed lines denote 95% confidence profiles.

Figure 4

Results obtained for system $E$, i.e., when two mutually anticorrelated groups of correlated time series are generated each of which contains two time series. (a) shows the time evolution of the largest eigenvalue ${\lambda }_{20}$ and (b) the symmetry parameter $S_{20}$ of the largest state. In (c) the results for the three smallest eigenvalues are shown: ${\lambda }_1$ (solid line), ${\lambda }_2$ (dashed line), and ${\lambda }_3$ (dotted line). In (d) the results for the symmetry parameter $S_i$ for the three smallest states are shown, using the same line coding as in (c). The time scale is given in arbitrary units as explained in the text.

Figure 5

Results obtained for system $F$, two mutually noncorrelated groups of correlated time series are generated. (a) shows the time evolution of the largest eigenvalue ${\lambda }_{20}$ (solid line) and the second largest eigenvalue (dashed line). In (b) the symmetry parameters $S_{20}$ of the largest state (solid line) and $S_{19}$ of the second largest eigenvector (dashed line) are shown. In (c) the results for the two smallest eigenvalues are shown: ${\lambda }_1$ (solid line), ${\lambda }_2$ (dashed line). In (d) the results for the symmetry parameter $S_i$ for the two smallest states are shown, using the same line coding as in (c). The time scale is given in arbitrary units as explained in the text.

Figure 6

Time development of (a) the largest eigenvalue ${\lambda }_{20}$ and (b) the smallest eigenvalue ${\lambda }_1$ for the MAR model Eq. (7), when only two from $M=20$ time series are correlated via the noise mixing described by Eq. (9) with $\rho =0.5$. The time scale is given in arbitrary units as explained in the text.

Figure 7

Eigenvalues as a function of $\rho$ for the MAR model Eq. (7), for the case when (a) only two from $M=20$ time series are correlated via noise mixing described by Eq. (9), and (b) two mutually uncorrelated groups are generated, each group containing two time series. The figure displays the average value taken over 100 trials for each ρ (solid lines) and 95% confidence profiles (dashed lines). In (a) the two largest and the two smallest eigenvalues are shown as functions of the mixing parameter $\rho$ [see Eq. (9)]. (b) shows the same for the three largest and three smallest eigenvalues.

Figure 8

$\eta$ dependence of the eigenvalues of the correlation matrix calculated from time series of the coupled Rössler system Eq. (11). An average over 100 trials for each value of η with different initial conditions is shown (solid line). Dashed lines denote 95% confidence profiles; (a) the two eigenvalues for $M=2$ coupled Rössler systems; (b) the two smallest eigenvalues when only two from a total of $M=20$ systems [Eq. (11)] are coupled.

Figure 9

(a) A segment of $130\mathrm{s}$ from an EEG recording at electrode Pz. From about $510\text{to}522\mathrm{s}$ the large amplitude activity of a generalized epileptic seizure can be seen. The inset magnifies the seizure activity. The time development of (b) the largest eigenvalue ${\lambda }_{19}$ and (c) the smallest eigenvalue of the correlation matrix are shown.